I'm sure there are other examples that non-algebraists would think of before these. All the examples I cited take the basic form of having several infinite families and some number of exceptional examples which do not fall into any of these families. Thus I am wondering:

Question: Does anyone know of examples of classifications of some mathematical objects such that the classification consists (A) only of infinite families or (B) only of a finite number of examples/ an infinite number of examples which do not seem to be closely related to one another (i.e. they do not "appear" to form any infinite families).

One example of case (B) that comes to mind would be finite dimensional division algebras over $\mathbb{R}$ of which there are 4. On the other hand, for case (B) I would like to rule out way too specific "classifications" such as "finite simple groups with an involution centralizer of such and such a form" since this is really a subclassification within the classification of FSG's. Although I am an algebraist, I would like to hear about examples from any branch of math, for comparison's sake.

Often the representations of some algebraic object that you care about can be described in some uniform way. Say the polynomial representations of $GL_n(\mathbb{C})$ or finite-dimensional $\mathfrak{sl}_2$ modules.
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Sam HopkinsJun 16 '14 at 20:05

10 Answers
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This feels like a somewhat silly example, but what about the classification of two (real) dimensional manifolds? They are two families of these, the orientable and non-orientable families classified by their genus/Euler characteristic.

I should add that this is an example of (A).
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Simon RoseMar 24 '11 at 18:53

Wow, I guess I totally overlooked such an obvious example from undergrad/grad topology classes.
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ARupinskiMar 24 '11 at 18:53

@Simon: I'm not really sure that this is an example of (A). I think one could argue that that orientable surfaces with genera $g>1$ are the ones making up the generic infinite family, whereas $g=0$ and $g=1$ are the two "sporadic" examples. It is just that in this case, the sporadic examples came first.
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José Figueroa-O'FarrillMar 25 '11 at 1:18

How about classification of irreducible closed 2-manifolds: just $S^2, T^2, and RP^2$. All others are (connected) sums of these. So maybe this is an example of (B). @Jose why do you view g=1 as "sporadic"? (hyperbolic metric, presumably?) topologically they aren't much more special than their higher genus siblings.
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PaulMar 25 '11 at 2:48

@José - I think it depends on how you view the classification. Purely topologically, I would argue that the two families are: $A_g$ = the connected sum of $g$ copies of $T^2$, and $B_k$ = the connected sum of $k$ copies of $RP^2$. Ignoring the geometry, this is perfectly valid. The only dodgy part of it, perhaps, is that $B_0 = A_0$, but this sort of thing also occurs in the classification of Lie Algebras as well, so...
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Simon RoseMar 25 '11 at 14:51

String theories. For example, if you confine yourself to bosonic string theory you find it only works in dimension 26 despite the definition being completely independent of the number of dimensions. Similarly there are just 5 superstring theories. All of these theories are closely related to other interesting classifications in mathematics.

As an example of (B), I'd mention Connes's classification of injective (type $II_{1}$) factors from the theory of von Neumann algebras. In this case, many apparently disparate constructions turn out to give a single object.

Andrews and Li (http://arxiv.org/pdf/1204.5007v3.pdf) have recently classified all embedded CMC (constant mean curvature) tori in the 3-sphere (up to isometries of $S^3$)
by using methods from Brendles proof of the Lawson conjecture:

If one does not specify the (mean curvature) constant $H$, the space of embedded
CMC tori is connected, but consist of infinitely many 1-dimensional families: The "exceptional" family consists of homogeneous CMC tori (parametrized by the mean curvature) which are (by definition) invariant under a 2-dimensional
group of spherical isometries. At certain values of $H$ (namely $H=cot(\frac{\pi}{m})$) there bifurcates of another 1-dimensional family of CMC tori whose symmetry group is $S^1\times \mathbb Z_m.$

In extensions of number fields $E/k$, you can look at primes in the ring of integers of $k$ that either (a) split in the integers of $E$, (b) remain prime in the integers of $E$, or (c) ramify. The case (c) consists of finitely many exceptions.

There are a lot of arguments in Diophantine equations using primes congruent to $1 \mod{6}$, primes congruent to $5 \mod{6}$, and two sporadic elements. Or you can have collections of families $\mod{n}$, with sporadic prime divisors of $n$. Maybe the most common: "Let $p$ be in the odd prime family..."
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Zack WolskeJun 16 '14 at 17:12